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Problem 7 Medium Difficulty

The average lifetime of a pi meson in its own frame of reference (i.e., the proper lifetime) is $2.6 \times 10^{-8} s.$ If the meson moves with a speed of 0.98$c,$ what is (a) its mean lifetime as measured by an observer on Earth, and (b) the average distance it travels before decaying, as measured by an observer on Earth? (c) What distance would it travel if time dilation did not occur?

Answer

a) 70 beats/min
b) 38$m$
c) 7.6$m$

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Video Transcript

But the proper lifetime by Maison is 2.6 multiplied by 10 to the hour. Eight seconds. Uh, appropriate lifetime. Inappropriate. The proper lifetime Lifetime. Oh, uh, by me zone. Ah, prime is on me. So on, uh, is a cool too. 2.6 months of life attend to the power minus eight seconds. 2.6. Multiply by 10 to the power minus eight seconds and speed of the maison is given. Ah, speed off Amazon. We is a cool too 0.98 times the speed of light. Uh, in part, it we have to find out mean lifetime is measured by an observer on earth. So using, uh, Delta T is equal to Delta tp divided by, uh, square root off one minus. We squared, divided by C square. Uh, we have appropriate time. Proper time is equal to 2.6 multiplied by 10 to the power minus eight seconds. Divided by, uh, squared. Who does? One minus. Well, we is given, which is 0.98 times the speed of light. And we have squared Israel divided by C square. See, school gets canceled with C square, and we have ah 2.6 Multiply by 10 to the power minus eight. Divided by ah, square root off zero point 03960396 Yeah. And therefore, Delta T is equal to 1.31 Multiply by 10 to the power, Uh, minus seven seconds. Minus seven seconds. Now let's do a Barbie. We know it. Uh, yes. Equals. We multiply by t in Barbie. We need to find out the average distance the maison trembles before became is measured by an observer on art. Therefore, using as physical to the amount of liberty um so s equals. We is Ah, cedar 0.98 times the speed off light multiply by t. Um, well, he is one point tree one. Multiply by 10 to the power minus seven. And therefore yes is equal to 38 point 51 meters. 51 meters. Now let's do part. See? Well, what distance, Woody travel If time dilation did more talker. Oh, this is the question. So again, using as physical to re multiply my tea Well, we is 0.90 times the speed of light. So if is equal to um 0.98 times The speed of light multiplied by T and T in this case is ah, 2.6 multiplied by 10 to the power minus eight seconds and multiplying. Uh, we have distance is equal to huh? Seven point fix meters.

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